A new pyramid-like shape always lands the same side up

Worst D-4 ever! But more seriously, I wonder how closely you could get to an non-uniform mass polyhedra which had 'knife edge' type balance. Which is to say;

1) Construct a polyhedra with uneven weight distribution which is stable on exactly two faces.

2) Make one of those faces much more stable than the other, so if it is on the limited stability face and disturbed, it will switch to the high stability face.

A structure like that would be useful as a tamper detector.

A solid tall cone is quite similar to what you want. I guess it can be tweaked to get a polyhedra.